Solve the Inequality for x 2x^2+4x+22<-10x+2

$2 x^{2} + 4 x + 22 < - 10 x + 2$

Move all terms containing $x$ to the left side of the inequality.

Add $10 x$ to both sides of the inequality.

$2 x^{2} + 4 x + 22 + 10 x < 2$

Add $4 x$ and $10 x$ .

$2 x^{2} + 14 x + 22 < 2$

Convert the inequality to an equation.

$2 x^{2} + 14 x + 22 = 2$

Subtract $2$ from both sides of the equation.

$2 x^{2} + 14 x + 22 - 2 = 0$

Subtract $2$ from $22$ .

$2 x^{2} + 14 x + 20 = 0$

Factor the left side of the equation.

Factor $2$ out of $2 x^{2} + 14 x + 20$ .

Factor $2$ out of $2 x^{2}$ .

$2 (x^{2}) + 14 x + 20 = 0$

Factor $2$ out of $14 x$ .

$2 (x^{2}) + 2 (7 x) + 20 = 0$

Factor $2$ out of $20$ .

$2 x^{2} + 2 (7 x) + 2 \cdot 10 = 0$

Factor $2$ out of $2 x^{2} + 2 (7 x)$ .

$2 (x^{2} + 7 x) + 2 \cdot 10 = 0$

Factor $2$ out of $2 (x^{2} + 7 x) + 2 \cdot 10$ .

$2 (x^{2} + 7 x + 10) = 0$

Factor.

Factor $x^{2} + 7 x + 10$ using the AC method.

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $10$ and whose sum is $7$ .

$2, 5$

Write the factored form using these integers.

$2 ((x + 2) (x + 5)) = 0$

Remove unnecessary parentheses.

$2 (x + 2) (x + 5) = 0$

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x + 2 = 0$

$x + 5 = 0$

Set $x + 2$ equal to $0$ and solve for $x$ .

Set $x + 2$ equal to $0$ .

$x + 2 = 0$

Subtract $2$ from both sides of the equation.

$x = - 2$

Set $x + 5$ equal to $0$ and solve for $x$ .

Set $x + 5$ equal to $0$ .

$x + 5 = 0$

Subtract $5$ from both sides of the equation.

$x = - 5$

The final solution is all the values that make $2 (x + 2) (x + 5) = 0$ true.

$x = - 2, - 5$

Use each root to create test intervals.

$x < - 5$

$- 5 < x < - 2$

$x > - 2$

Choose a test value from each interval and plug this value into the original inequality to determine which intervals satisfy the inequality.

Test a value on the interval $x < - 5$ to see if it makes the inequality true.

Choose a value on the interval $x < - 5$ and see if this value makes the original inequality true.

$x = - 8$

Replace $x$ with $- 8$ in the original inequality.

$2 {(- 8)}^{2} + 4 (- 8) + 22 < - 10 \cdot - 8 + 2$

The left side $118$ is not less than the right side $82$ , which means that the given statement is false.

False

Test a value on the interval $- 5 < x < - 2$ to see if it makes the inequality true.

Choose a value on the interval $- 5 < x < - 2$ and see if this value makes the original inequality true.

$x = - 4$

Replace $x$ with $- 4$ in the original inequality.

$2 {(- 4)}^{2} + 4 (- 4) + 22 < - 10 \cdot - 4 + 2$

The left side $38$ is less than the right side $42$ , which means that the given statement is always true.

True

Test a value on the interval $x > - 2$ to see if it makes the inequality true.

Choose a value on the interval $x > - 2$ and see if this value makes the original inequality true.

$x = 0$

Replace $x$ with $0$ in the original inequality.

$2 {(0)}^{2} + 4 (0) + 22 < - 10 \cdot 0 + 2$

The left side $22$ is not less than the right side $2$ , which means that the given statement is false.

False

Compare the intervals to determine which ones satisfy the original inequality.

$x < - 5$ False

$- 5 < x < - 2$ True

$x > - 2$ False

$x < - 5$ False

$- 5 < x < - 2$ True

$x > - 2$ False

The solution consists of all of the true intervals.

$- 5 < x < - 2$

The result can be shown in multiple forms.

Inequality Form:

$- 5 < x < - 2$

Interval Notation:

$(- 5, - 2)$

Solve the Inequality for x 2x^2+4x+22<-10x+2

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